P Only If Q

But that is exactly what P Q rules out. In other words ptrue and qfalse is the only invalid combination.

This Was An Actual Question In A Paper Set By The Indian Statistical Institute This Or That Questions Interesting Questions Let It Be

Salt Lake City is in Utah Q.

P only if q. Inverse Statement-If you will not qualify GATE then you do not work hard. This means that whenever we have p it must also be that we have q as p can happen only if we have q. P only if q q -- p p -- q p if q q -- p p is sufficient for q p -- q p is necessary for q q -- p p is sufficient and necessary p q for q ie.

Pqpr p or q and r is equiv. Name Represented Meaning Negation p not p Conjunction pq p and q Disjunction pq p or q or both Exclusive Or pq either p or q but not both Implication p q. Otherwise it is true.

P is a sufficient condition for q means if p then q Necessary condition. P Q P Q T T T T F F F T F F F F Note that the conjunction P Q is true only when both P and Q are true. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.

_Biconditional _ statements are conditional statements that are true if the statement is still true when the antecedent and consequent are reversed. P if and only if Q is rarely found in ordinary English. Of q by p is If p then q or p implies q We symbolize the conditional by p q and frequently read p implies q or It is false when p is true and q is false.

To if not q then not p Material Implication p q pq if p then q is equiv. The biconditional of p and q is p if and only if q and is denoted p q. Suppose pq is true and pqis falsepq would be true if pqis false.

That is P only if Q rules out just one possibility. Given two statements P Q the compound statement P and Q called the conjunction is denoted by P Q and is defined by the following truth table. AIf it is hot outside you buy an ice cream cone and if you buy an ice cream cone it is hot.

We have settled the semantics for if and only if. The Biconditional Operator The Biconditional Operator The biconditional operator is used to represent a two-directional implication. Q follows from p.

Pqis true only if pis true and qis false. The main ones are the following p and q represent given propositions. Suppose pq is false and pqis true.

This is logically equivalent to p implies q which is valid if pqtrue or pfalse. If p then q. Q is necessary and sufficient for P P is equivalent or materially equivalent to Q compare with material implication P precisely if Q P precisely or exactly when Q P.

Let p and q are two statements then if p then q is a compound statement denoted by p q and referred as a conditional statement or implication. In fact when P if and only Q is true P can subsitute for Q and Q can subsitute for P in other compound sentences without changing the truth. What should its truth table look like.

P only if q. But in this case pq will be true. 14 25 Write each of these propositions in the form p if and only if q in English.

Otherwise it is always true. Converse Statement-If you work hard then you will qualify GATE. This sentence is of the form- p only if q.

You work hard. Its rather legalistic sounding. You will qualify GATE.

However if pis true and qis false then pqwill be true. That P is true and Q is false. This is the same as p only if q again because this only excludes the ptrue qfalse case.

By asserting an implication one asserts that it does not occur that the antecedent is true and the consequent is false. We can now introduce a new symbol for this expression. Hence this case is not possible.

The implication p q is false only when p is true and q is false. Proposition q is necessary for p means that when p is true it must result from q being true. So its obviously correct to read P Q as P only if Q.

If this is unclear to you go back and review section 22 Now let us make a truth table for this formula. P if Q means that the truth of Q is sufficient or enough for P. In writing phrases commonly used as alternatives to P if and only if Q include.

P only if q means that q is a necessary condition for p. Contrapositive Statement-If you do not work hard then you will not qualify GATE. Q is necessary for p.

If on the other hand introduces a sufficient condition. Specifically P if Q and P only if Q will be translated QP PQ. To p or q and p or r Double Negation p p p is equivalent to the negation of not p Transposition p q q p if p then q is equiv.

P is sufcient for q. Discrete Mathematics I Fall 2014 11 pg. Specifically p qmeans that pimplies q and qimplies p.

The only way for p q to be false is for p to be true and q to be false. So the symbolic form is p q where-p. Example Proposition p.

The compound statement p. If P Q are the statements P. An implication is thus.

Now this only occurs if pis true and qis false. Think about it. It is traditional to use the double arrow.

P only if Q means that the truth of Q is necessary or required in order for P to be true. P if and only if q The best approach in doing a proof by contrapositive is to restate the original problem in the form If p then q. C Xin He University at Buffalo CSE 191 Discrete Structures 15 37 Terminology for implication.

It means that p can occur only when q has occurred. When P if and only if Q is true it is often said that P and Q are logically equivalent. To not p or q Exportation pq r p q r.

Otherwise pqis by definitiontrue. Q ifp. P implies q.

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